Free normalization retained from X-G-ST

Set

and use positive retarded/advanced inverses ${{\mathsf{E}}_0}^{\mathrm{R}},{{\mathsf{E}}_0}^{\mathrm{A}}$ of $\square$ and ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ of $P_m$. They are used only on the completion BGP.

For covariant symmetric tensors and vector fields define

With the tensor–density pairing,

In particular,

The fields, parities, and ghost numbers are

The independent auxiliary field is retained. The free differential and gauge fermion are

Thus

Eliminating $b$ gives $b=\mathsf{C}r/2$ and the Feynman weight $(\mathsf{C}r{)}^2/4$.

For reference, the retained Green blocks are

The scalar Hessian inverse is $-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}/\mathrm{A}}$, not the positive equation inverse.

These matrices invert, respectively,

For the first block this follows by substituting $b=(\mathsf{C}r-j)/2$ and solving ${\mathsf{P}}_{\mathrm{gr}}r=J+{\mathsf{C}}^{\ast }j/2$. For the ghost block it follows by matrix multiplication. This records the normalization without constructing a new free theory XG,XGST.

The free causal algebra ${\mathfrak{A}}_0$ is the graded polynomial algebra of compactly smeared linear fields with these causal relations and the equations

Its equation ideal is denoted ${\mathcal{J}}_0$. X-G-ST's nilpotent causal derivation $s_0$ is retained. No interacting action on ${\mathfrak{A}}_0$ follows from that free result.